(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_filter.ML

     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

     Author:     Jasmin Blanchette, TU Muenchen

 Sledgehammer's relevance filter.

 *)

 signature SLEDGEHAMMER_FILTER =

 sig

   type status = ATP_Problem_Generate.status

   type stature = ATP_Problem_Generate.stature

   type relevance_fudge =

     {local_const_multiplier : real,

      worse_irrel_freq : real,

      higher_order_irrel_weight : real,

      abs_rel_weight : real,

      abs_irrel_weight : real,

      skolem_irrel_weight : real,

      theory_const_rel_weight : real,

      theory_const_irrel_weight : real,

      chained_const_irrel_weight : real,

      intro_bonus : real,

      elim_bonus : real,

      simp_bonus : real,

      local_bonus : real,

      assum_bonus : real,

      chained_bonus : real,

      max_imperfect : real,

      max_imperfect_exp : real,

      threshold_divisor : real,

      ridiculous_threshold : real}

   type relevance_override =

     {add : (Facts.ref * Attrib.src list) list,

      del : (Facts.ref * Attrib.src list) list,

      only : bool}

   val trace : bool Config.T

   val ignore_no_atp : bool Config.T

   val instantiate_inducts : bool Config.T

   val no_relevance_override : relevance_override

   val const_names_in_fact :

     theory -> (string * typ -> term list -> bool * term list) -> term

     -> string list

   val clasimpset_rule_table_of : Proof.context -> status Termtab.table

   val fact_from_ref :

     Proof.context -> unit Symtab.table -> thm list -> status Termtab.table

     -> Facts.ref * Attrib.src list -> ((string * stature) * thm) list

   val all_facts :

     Proof.context -> bool -> 'a Symtab.table -> bool -> thm list

     -> thm list -> status Termtab.table -> (((unit -> string) * stature) * thm) list

   val nearly_all_facts :

     Proof.context -> bool -> relevance_override -> thm list -> term list -> term

     -> (((unit -> string) * stature) * thm) list

   val relevant_facts :

     Proof.context -> real * real -> int

     -> (string * typ -> term list -> bool * term list) -> relevance_fudge

     -> relevance_override -> thm list -> term list -> term

     -> (((unit -> string) * stature) * thm) list

     -> ((string * stature) * thm) list

 end;

 structure Sledgehammer_Filter : SLEDGEHAMMER_FILTER =

 struct

 open ATP_Problem_Generate

 open Metis_Tactic

 open Sledgehammer_Util

 val trace =

   Attrib.setup_config_bool @{binding sledgehammer_filter_trace} (K false)

 fun trace_msg ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ()

 (* experimental features *)

 val ignore_no_atp =

   Attrib.setup_config_bool @{binding sledgehammer_ignore_no_atp} (K false)

 val instantiate_inducts =

   Attrib.setup_config_bool @{binding sledgehammer_instantiate_inducts} (K false)

 type relevance_fudge =

   {local_const_multiplier : real,

    worse_irrel_freq : real,

    higher_order_irrel_weight : real,

    abs_rel_weight : real,

    abs_irrel_weight : real,

    skolem_irrel_weight : real,

    theory_const_rel_weight : real,

    theory_const_irrel_weight : real,

    chained_const_irrel_weight : real,

    intro_bonus : real,

    elim_bonus : real,

    simp_bonus : real,

    local_bonus : real,

    assum_bonus : real,

    chained_bonus : real,

    max_imperfect : real,

    max_imperfect_exp : real,

    threshold_divisor : real,

    ridiculous_threshold : real}

 type relevance_override =

   {add : (Facts.ref * Attrib.src list) list,

    del : (Facts.ref * Attrib.src list) list,

    only : bool}

 val no_relevance_override = {add = [], del = [], only = false}

 val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator

 val abs_name = sledgehammer_prefix ^ "abs"

 val skolem_prefix = sledgehammer_prefix ^ "sko"

 val theory_const_suffix = Long_Name.separator ^ " 1"

 (* unfolding these can yield really huge terms *)

 val risky_spec_eqs = @{thms Bit0_def Bit1_def}

 fun clasimpset_rule_table_of ctxt =

   let

     val thy = Proof_Context.theory_of ctxt

     val atomize = HOLogic.mk_Trueprop o Object_Logic.atomize_term thy

     fun add stature normalizers get_th =

       fold (fn rule =>

                let

                  val th = rule |> get_th

                  val t =

                    th |> Thm.maxidx_of th > 0 ? zero_var_indexes |> prop_of

                in

                  fold (fn normalize => Termtab.update (normalize t, stature))

                       (I :: normalizers)

                end)

 (*

     TODO: "intro" and "elim" rules are not exploited yet by SPASS (or any other

     prover). Reintroduce this code when it becomes useful. It costs about 50 ms

     per Sledgehammer invocation.

     val {safeIs, safeEs, hazIs, hazEs, ...} =

       ctxt |> claset_of |> Classical.rep_cs

     val intros = Item_Net.content safeIs @ Item_Net.content hazIs

     val elims =

       Item_Net.content safeEs @ Item_Net.content hazEs

       |> map Classical.classical_rule

 *)

     val simps = ctxt |> simpset_of |> dest_ss |> #simps

     val spec_eqs =

       ctxt |> Spec_Rules.get

            |> filter (curry (op =) Spec_Rules.Equational o fst)

            |> maps (snd o snd)

            |> filter_out (member Thm.eq_thm_prop risky_spec_eqs)

   in

     Termtab.empty |> add Simp [atomize] snd simps

                   |> add Spec_Eq [] I spec_eqs

 (*

                   |> add Elim [] I elims

                   |> add Intro [] I intros

 *)

   end

 fun needs_quoting reserved s =

   Symtab.defined reserved s orelse

   exists (not o Lexicon.is_identifier) (Long_Name.explode s)

 fun make_name reserved multi j name =

   (name |> needs_quoting reserved name ? quote) ^

   (if multi then "(" ^ string_of_int j ^ ")" else "")

 fun explode_interval _ (Facts.FromTo (i, j)) = i upto j

   | explode_interval max (Facts.From i) = i upto i + max - 1

   | explode_interval _ (Facts.Single i) = [i]

 val backquote =

   raw_explode #> map (fn "`" => "\\`" | s => s) #> implode #> enclose "`" "`"

 fun is_assum assms th = exists (fn ct => prop_of th aconv term_of ct) assms

 fun is_chained chained_ths = member Thm.eq_thm_prop chained_ths

 fun scope_of_thm global assms chained_ths th =

   if is_chained chained_ths th then Chained

   else if global then Global

   else if is_assum assms th then Assum

   else Local

 fun status_of_thm css_table name th =

   (* FIXME: use structured name *)

   if String.isSubstring ".induct" name orelse

      String.isSubstring ".inducts" name then

     Induct

   else case Termtab.lookup css_table (prop_of th) of

     SOME status => status

   | NONE => General

 fun stature_of_thm global assms chained_ths css_table name th =

   (scope_of_thm global assms chained_ths th, status_of_thm css_table name th)

 fun fact_from_ref ctxt reserved chained_ths css_table (xthm as (xref, args)) =

   let

     val ths = Attrib.eval_thms ctxt [xthm]

     val bracket =

       map (enclose "[" "]" o Pretty.str_of o Args.pretty_src ctxt) args

       |> implode

     fun nth_name j =

       case xref of

         Facts.Fact s => backquote s ^ bracket

       | Facts.Named (("", _), _) => "[" ^ bracket ^ "]"

       | Facts.Named ((name, _), NONE) =>

         make_name reserved (length ths > 1) (j + 1) name ^ bracket

       | Facts.Named ((name, _), SOME intervals) =>

         make_name reserved true

                  (nth (maps (explode_interval (length ths)) intervals) j) name ^

         bracket

   in

     (ths, (0, []))

     |-> fold (fn th => fn (j, rest) =>

                  let val name = nth_name j in

                    (j + 1, ((name, stature_of_thm false [] chained_ths

                                              css_table name th), th) :: rest)

                  end)

     |> snd

   end

 (* This is a terrible hack. Free variables are sometimes coded as "M__" when

    they are displayed as "M" and we want to avoid clashes with these. But

    sometimes it's even worse: "Ma__" encodes "M". So we simply reserve all

    prefixes of all free variables. In the worse case scenario, where the fact

    won't be resolved correctly, the user can fix it manually, e.g., by naming

    the fact in question. Ideally we would need nothing of it, but backticks

    simply don't work with schematic variables. *)

 fun all_prefixes_of s =

   map (fn i => String.extract (s, 0, SOME i)) (1 upto size s - 1)

 fun close_form t =

   (t, [] |> Term.add_free_names t |> maps all_prefixes_of)

   |> fold (fn ((s, i), T) => fn (t', taken) =>

               let val s' = singleton (Name.variant_list taken) s in

                 ((if fastype_of t' = HOLogic.boolT then HOLogic.all_const

                   else Logic.all_const) T

                  $ Abs (s', T, abstract_over (Var ((s, i), T), t')),

                  s' :: taken)

               end)

           (Term.add_vars t [] |> sort_wrt (fst o fst))

   |> fst

 fun string_for_term ctxt t =

   Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)

                    (print_mode_value ())) (Syntax.string_of_term ctxt) t

   |> String.translate (fn c => if Char.isPrint c then str c else "")

   |> simplify_spaces

 (** Structural induction rules **)

 fun struct_induct_rule_on th =

   case Logic.strip_horn (prop_of th) of

     (prems, @{const Trueprop}

             $ ((p as Var ((p_name, 0), _)) $ (a as Var (_, ind_T)))) =>

     if not (is_TVar ind_T) andalso length prems > 1 andalso

        exists (exists_subterm (curry (op aconv) p)) prems andalso

        not (exists (exists_subterm (curry (op aconv) a)) prems) then

       SOME (p_name, ind_T)

     else

       NONE

   | _ => NONE

 fun instantiate_induct_rule ctxt concl_prop p_name ((name, stature), th) ind_x =

   let

     fun varify_noninducts (t as Free (s, T)) =

         if (s, T) = ind_x orelse can dest_funT T then t else Var ((s, 0), T)

       | varify_noninducts t = t

     val p_inst =

       concl_prop |> map_aterms varify_noninducts |> close_form

                  |> lambda (Free ind_x)

                  |> string_for_term ctxt

   in

     ((fn () => name () ^ "[where " ^ p_name ^ " = " ^ quote p_inst ^ "]",

       stature), th |> read_instantiate ctxt [((p_name, 0), p_inst)])

   end

 fun type_match thy (T1, T2) =

   (Sign.typ_match thy (T2, T1) Vartab.empty; true)

   handle Type.TYPE_MATCH => false

 fun instantiate_if_induct_rule ctxt stmt stmt_xs (ax as (_, th)) =

   case struct_induct_rule_on th of

     SOME (p_name, ind_T) =>

     let val thy = Proof_Context.theory_of ctxt in

       stmt_xs |> filter (fn (_, T) => type_match thy (T, ind_T))

               |> map_filter (try (instantiate_induct_rule ctxt stmt p_name ax))

     end

   | NONE => [ax]

 (***************************************************************)

 (* Relevance Filtering                                         *)

 (***************************************************************)

 (*** constants with types ***)

 fun order_of_type (Type (@{type_name fun}, [T1, T2])) =

     Int.max (order_of_type T1 + 1, order_of_type T2)

   | order_of_type (Type (_, Ts)) = fold (Integer.max o order_of_type) Ts 0

   | order_of_type _ = 0

 (* An abstraction of Isabelle types and first-order terms *)

 datatype pattern = PVar | PApp of string * pattern list

 datatype ptype = PType of int * pattern list

 fun string_for_pattern PVar = "_"

   | string_for_pattern (PApp (s, ps)) =

     if null ps then s else s ^ string_for_patterns ps

 and string_for_patterns ps = "(" ^ commas (map string_for_pattern ps) ^ ")"

 fun string_for_ptype (PType (_, ps)) = string_for_patterns ps

 (*Is the second type an instance of the first one?*)

 fun match_pattern (PVar, _) = true

   | match_pattern (PApp _, PVar) = false

   | match_pattern (PApp (s, ps), PApp (t, qs)) =

     s = t andalso match_patterns (ps, qs)

 and match_patterns (_, []) = true

   | match_patterns ([], _) = false

   | match_patterns (p :: ps, q :: qs) =

     match_pattern (p, q) andalso match_patterns (ps, qs)

 fun match_ptype (PType (_, ps), PType (_, qs)) = match_patterns (ps, qs)

 (* Is there a unifiable constant? *)

 fun pconst_mem f consts (s, ps) =

   exists (curry (match_ptype o f) ps)

          (map snd (filter (curry (op =) s o fst) consts))

 fun pconst_hyper_mem f const_tab (s, ps) =

   exists (curry (match_ptype o f) ps) (these (Symtab.lookup const_tab s))

 fun pattern_for_type (Type (s, Ts)) = PApp (s, map pattern_for_type Ts)

   | pattern_for_type (TFree (s, _)) = PApp (s, [])

   | pattern_for_type (TVar _) = PVar

 (* Pairs a constant with the list of its type instantiations. *)

 fun ptype thy const x =

   (if const then map pattern_for_type (these (try (Sign.const_typargs thy) x))

    else [])

 fun rich_ptype thy const (s, T) =

   PType (order_of_type T, ptype thy const (s, T))

 fun rich_pconst thy const (s, T) = (s, rich_ptype thy const (s, T))

 fun string_for_hyper_pconst (s, ps) =

   s ^ "{" ^ commas (map string_for_ptype ps) ^ "}"

 (* Add a pconstant to the table, but a [] entry means a standard

    connective, which we ignore.*)

 fun add_pconst_to_table also_skolem (s, p) =

   if (not also_skolem andalso String.isPrefix skolem_prefix s) then I

   else Symtab.map_default (s, [p]) (insert (op =) p)

 fun add_pconsts_in_term thy is_built_in_const also_skolems pos =

   let

     val flip = Option.map not

     (* We include free variables, as well as constants, to handle locales. For

        each quantifiers that must necessarily be skolemized by the automatic

        prover, we introduce a fresh constant to simulate the effect of

        Skolemization. *)

     fun do_const const x ts =

       let val (built_in, ts) = is_built_in_const x ts in

         (not built_in

          ? add_pconst_to_table also_skolems (rich_pconst thy const x))

         #> fold (do_term false) ts

       end

     and do_term ext_arg t =

       case strip_comb t of

         (Const x, ts) => do_const true x ts

       | (Free x, ts) => do_const false x ts

       | (Abs (_, T, t'), ts) =>

         ((null ts andalso not ext_arg)

          (* Since lambdas on the right-hand side of equalities are usually

             extensionalized later by "extensionalize_term", we don't penalize

             them here. *)

          ? add_pconst_to_table true (abs_name, PType (order_of_type T + 1, [])))

         #> fold (do_term false) (t' :: ts)

       | (_, ts) => fold (do_term false) ts

     fun do_quantifier will_surely_be_skolemized abs_T body_t =

       do_formula pos body_t

       #> (if also_skolems andalso will_surely_be_skolemized then

             add_pconst_to_table true

                 (legacy_gensym skolem_prefix, PType (order_of_type abs_T, []))

           else

             I)

     and do_term_or_formula ext_arg T =

       if T = HOLogic.boolT then do_formula NONE else do_term ext_arg

     and do_formula pos t =

       case t of

         Const (@{const_name all}, _) $ Abs (_, T, t') =>

         do_quantifier (pos = SOME false) T t'

       | @{const "==>"} $ t1 $ t2 =>

         do_formula (flip pos) t1 #> do_formula pos t2

       | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>

         do_term_or_formula false T t1 #> do_term_or_formula true T t2

       | @{const Trueprop} $ t1 => do_formula pos t1

       | @{const False} => I

       | @{const True} => I

       | @{const Not} $ t1 => do_formula (flip pos) t1

       | Const (@{const_name All}, _) $ Abs (_, T, t') =>

         do_quantifier (pos = SOME false) T t'

       | Const (@{const_name Ex}, _) $ Abs (_, T, t') =>

         do_quantifier (pos = SOME true) T t'

       | @{const HOL.conj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]

       | @{const HOL.disj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]

       | @{const HOL.implies} $ t1 $ t2 =>

         do_formula (flip pos) t1 #> do_formula pos t2

       | Const (@{const_name HOL.eq}, Type (_, [T, _])) $ t1 $ t2 =>

         do_term_or_formula false T t1 #> do_term_or_formula true T t2

       | Const (@{const_name If}, Type (_, [_, Type (_, [T, _])]))

         $ t1 $ t2 $ t3 =>

         do_formula NONE t1 #> fold (do_term_or_formula false T) [t2, t3]

       | Const (@{const_name Ex1}, _) $ Abs (_, T, t') =>

         do_quantifier (is_some pos) T t'

       | Const (@{const_name Ball}, _) $ t1 $ Abs (_, T, t') =>

         do_quantifier (pos = SOME false) T

                       (HOLogic.mk_imp (incr_boundvars 1 t1 $ Bound 0, t'))

       | Const (@{const_name Bex}, _) $ t1 $ Abs (_, T, t') =>

         do_quantifier (pos = SOME true) T

                       (HOLogic.mk_conj (incr_boundvars 1 t1 $ Bound 0, t'))

       | (t0 as Const (_, @{typ bool})) $ t1 =>

         do_term false t0 #> do_formula pos t1  (* theory constant *)

       | _ => do_term false t

   in do_formula pos end

 (*Inserts a dummy "constant" referring to the theory name, so that relevance

   takes the given theory into account.*)

 fun theory_constify ({theory_const_rel_weight, theory_const_irrel_weight, ...}

                      : relevance_fudge) thy_name t =

   if exists (curry (op <) 0.0) [theory_const_rel_weight,

                                 theory_const_irrel_weight] then

     Const (thy_name ^ theory_const_suffix, @{typ bool}) $ t

   else

     t

 fun theory_const_prop_of fudge th =

   theory_constify fudge (Context.theory_name (theory_of_thm th)) (prop_of th)

 (**** Constant / Type Frequencies ****)

 (* A two-dimensional symbol table counts frequencies of constants. It's keyed

    first by constant name and second by its list of type instantiations. For the

    latter, we need a linear ordering on "pattern list". *)

 fun pattern_ord p =

   case p of

     (PVar, PVar) => EQUAL

   | (PVar, PApp _) => LESS

   | (PApp _, PVar) => GREATER

   | (PApp q1, PApp q2) =>

     prod_ord fast_string_ord (dict_ord pattern_ord) (q1, q2)

 fun ptype_ord (PType p, PType q) =

   prod_ord (dict_ord pattern_ord) int_ord (swap p, swap q)

 structure PType_Tab = Table(type key = ptype val ord = ptype_ord)

 fun count_fact_consts thy fudge =

   let

     fun do_const const (s, T) ts =

       (* Two-dimensional table update. Constant maps to types maps to count. *)

       PType_Tab.map_default (rich_ptype thy const (s, T), 0) (Integer.add 1)

       |> Symtab.map_default (s, PType_Tab.empty)

       #> fold do_term ts

     and do_term t =

       case strip_comb t of

         (Const x, ts) => do_const true x ts

       | (Free x, ts) => do_const false x ts

       | (Abs (_, _, t'), ts) => fold do_term (t' :: ts)

       | (_, ts) => fold do_term ts

   in do_term o theory_const_prop_of fudge o snd end

 (**** Actual Filtering Code ****)

 fun pow_int _ 0 = 1.0

   | pow_int x 1 = x

   | pow_int x n = if n > 0 then x * pow_int x (n - 1) else pow_int x (n + 1) / x

 (*The frequency of a constant is the sum of those of all instances of its type.*)

 fun pconst_freq match const_tab (c, ps) =

   PType_Tab.fold (fn (qs, m) => match (ps, qs) ? Integer.add m)

                  (the (Symtab.lookup const_tab c)) 0

 (* A surprising number of theorems contain only a few significant constants.

    These include all induction rules, and other general theorems. *)

 (* "log" seems best in practice. A constant function of one ignores the constant

    frequencies. Rare constants give more points if they are relevant than less

    rare ones. *)

 fun rel_weight_for _ freq = 1.0 + 2.0 / Math.ln (Real.fromInt freq + 1.0)

 (* Irrelevant constants are treated differently. We associate lower penalties to

    very rare constants and very common ones -- the former because they can't

    lead to the inclusion of too many new facts, and the latter because they are

    so common as to be of little interest. *)

 fun irrel_weight_for ({worse_irrel_freq, higher_order_irrel_weight, ...}

                       : relevance_fudge) order freq =

   let val (k, x) = worse_irrel_freq |> `Real.ceil in

     (if freq < k then Math.ln (Real.fromInt (freq + 1)) / Math.ln x

      else rel_weight_for order freq / rel_weight_for order k)

     * pow_int higher_order_irrel_weight (order - 1)

   end

 fun multiplier_for_const_name local_const_multiplier s =

   if String.isSubstring "." s then 1.0 else local_const_multiplier

 (* Computes a constant's weight, as determined by its frequency. *)

 fun generic_pconst_weight local_const_multiplier abs_weight skolem_weight

                           theory_const_weight chained_const_weight weight_for f

                           const_tab chained_const_tab (c as (s, PType (m, _))) =

   if s = abs_name then

     abs_weight

   else if String.isPrefix skolem_prefix s then

     skolem_weight

   else if String.isSuffix theory_const_suffix s then

     theory_const_weight

   else

     multiplier_for_const_name local_const_multiplier s

     * weight_for m (pconst_freq (match_ptype o f) const_tab c)

     |> (if chained_const_weight < 1.0 andalso

            pconst_hyper_mem I chained_const_tab c then

           curry (op *) chained_const_weight

         else

           I)

 fun rel_pconst_weight ({local_const_multiplier, abs_rel_weight,

                         theory_const_rel_weight, ...} : relevance_fudge)

                       const_tab =

   generic_pconst_weight local_const_multiplier abs_rel_weight 0.0

                         theory_const_rel_weight 0.0 rel_weight_for I const_tab

                         Symtab.empty

 fun irrel_pconst_weight (fudge as {local_const_multiplier, abs_irrel_weight,

                                    skolem_irrel_weight,

                                    theory_const_irrel_weight,

                                    chained_const_irrel_weight, ...})

                         const_tab chained_const_tab =

   generic_pconst_weight local_const_multiplier abs_irrel_weight

                         skolem_irrel_weight theory_const_irrel_weight

                         chained_const_irrel_weight (irrel_weight_for fudge) swap

                         const_tab chained_const_tab

 fun stature_bonus ({intro_bonus, ...} : relevance_fudge) (_, Intro) =

     intro_bonus

   | stature_bonus {elim_bonus, ...} (_, Elim) = elim_bonus

   | stature_bonus {simp_bonus, ...} (_, Simp) = simp_bonus

   | stature_bonus {local_bonus, ...} (Local, _) = local_bonus

   | stature_bonus {assum_bonus, ...} (Assum, _) = assum_bonus

   | stature_bonus {chained_bonus, ...} (Chained, _) = chained_bonus

   | stature_bonus _ _ = 0.0

 fun is_odd_const_name s =

   s = abs_name orelse String.isPrefix skolem_prefix s orelse

   String.isSuffix theory_const_suffix s

 fun fact_weight fudge stature const_tab relevant_consts chained_consts

                 fact_consts =

   case fact_consts |> List.partition (pconst_hyper_mem I relevant_consts)

                    ||> filter_out (pconst_hyper_mem swap relevant_consts) of

     ([], _) => 0.0

   | (rel, irrel) =>

     if forall (forall (is_odd_const_name o fst)) [rel, irrel] then

       0.0

     else

       let

         val irrel = irrel |> filter_out (pconst_mem swap rel)

         val rel_weight =

           0.0 |> fold (curry (op +) o rel_pconst_weight fudge const_tab) rel

         val irrel_weight =

           ~ (stature_bonus fudge stature)

           |> fold (curry (op +)

                    o irrel_pconst_weight fudge const_tab chained_consts) irrel

         val res = rel_weight / (rel_weight + irrel_weight)

       in if Real.isFinite res then res else 0.0 end

 fun pconsts_in_fact thy is_built_in_const t =

   Symtab.fold (fn (s, pss) => fold (cons o pair s) pss)

               (Symtab.empty |> add_pconsts_in_term thy is_built_in_const true

                                                    (SOME true) t) []

 fun pair_consts_fact thy is_built_in_const fudge fact =

   case fact |> snd |> theory_const_prop_of fudge

             |> pconsts_in_fact thy is_built_in_const of

     [] => NONE

   | consts => SOME ((fact, consts), NONE)

 val const_names_in_fact = map fst ooo pconsts_in_fact

 type annotated_thm =

   (((unit -> string) * stature) * thm) * (string * ptype) list

 fun take_most_relevant ctxt max_relevant remaining_max

         ({max_imperfect, max_imperfect_exp, ...} : relevance_fudge)

         (candidates : (annotated_thm * real) list) =

   let

     val max_imperfect =

       Real.ceil (Math.pow (max_imperfect,

                     Math.pow (Real.fromInt remaining_max

                               / Real.fromInt max_relevant, max_imperfect_exp)))

     val (perfect, imperfect) =

       candidates |> sort (Real.compare o swap o pairself snd)

                  |> take_prefix (fn (_, w) => w > 0.99999)

     val ((accepts, more_rejects), rejects) =

       chop max_imperfect imperfect |>> append perfect |>> chop remaining_max

   in

     trace_msg ctxt (fn () =>

         "Actually passed (" ^ string_of_int (length accepts) ^ " of " ^

         string_of_int (length candidates) ^ "): " ^

         (accepts |> map (fn ((((name, _), _), _), weight) =>

                             name () ^ " [" ^ Real.toString weight ^ "]")

                  |> commas));

     (accepts, more_rejects @ rejects)

   end

 fun if_empty_replace_with_scope thy is_built_in_const facts sc tab =

   if Symtab.is_empty tab then

     Symtab.empty

     |> fold (add_pconsts_in_term thy is_built_in_const false (SOME false))

             (map_filter (fn ((_, (sc', _)), th) =>

                             if sc' = sc then SOME (prop_of th) else NONE) facts)

   else

     tab

 fun consider_arities is_built_in_const th =

   let

     fun aux _ _ NONE = NONE

       | aux t args (SOME tab) =

         case t of

           t1 $ t2 => SOME tab |> aux t1 (t2 :: args) |> aux t2 []

         | Const (x as (s, _)) =>

           (if is_built_in_const x args |> fst then

              SOME tab

            else case Symtab.lookup tab s of

              NONE => SOME (Symtab.update (s, length args) tab)

            | SOME n => if n = length args then SOME tab else NONE)

         | _ => SOME tab

   in aux (prop_of th) [] end

 (* FIXME: This is currently only useful for polymorphic type encodings. *)

 fun could_benefit_from_ext is_built_in_const facts =

   fold (consider_arities is_built_in_const o snd) facts (SOME Symtab.empty)

   |> is_none

 (* High enough so that it isn't wrongly considered as very relevant (e.g., for E

    weights), but low enough so that it is unlikely to be truncated away if few

    facts are included. *)

 val special_fact_index = 75

 fun relevance_filter ctxt threshold0 decay max_relevant is_built_in_const

         (fudge as {threshold_divisor, ridiculous_threshold, ...})

         ({add, del, ...} : relevance_override) facts chained_ts hyp_ts concl_t =

   let

     val thy = Proof_Context.theory_of ctxt

     val const_tab = fold (count_fact_consts thy fudge) facts Symtab.empty

     val add_pconsts = add_pconsts_in_term thy is_built_in_const false o SOME

     val chained_const_tab = Symtab.empty |> fold (add_pconsts true) chained_ts

     val goal_const_tab =

       Symtab.empty |> fold (add_pconsts true) hyp_ts

                    |> add_pconsts false concl_t

       |> (fn tab => if Symtab.is_empty tab then chained_const_tab else tab)

       |> fold (if_empty_replace_with_scope thy is_built_in_const facts)

               [Chained, Assum, Local]

     val add_ths = Attrib.eval_thms ctxt add

     val del_ths = Attrib.eval_thms ctxt del

     val facts = facts |> filter_out (member Thm.eq_thm_prop del_ths o snd)

     fun iter j remaining_max threshold rel_const_tab hopeless hopeful =

       let

         fun relevant [] _ [] =

             (* Nothing has been added this iteration. *)

             if j = 0 andalso threshold >= ridiculous_threshold then

               (* First iteration? Try again. *)

               iter 0 max_relevant (threshold / threshold_divisor) rel_const_tab

                    hopeless hopeful

             else

               []

           | relevant candidates rejects [] =

             let

               val (accepts, more_rejects) =

                 take_most_relevant ctxt max_relevant remaining_max fudge

                                    candidates

               val rel_const_tab' =

                 rel_const_tab

                 |> fold (add_pconst_to_table false) (maps (snd o fst) accepts)

               fun is_dirty (c, _) =

                 Symtab.lookup rel_const_tab' c <> Symtab.lookup rel_const_tab c

               val (hopeful_rejects, hopeless_rejects) =

                  (rejects @ hopeless, ([], []))

                  |-> fold (fn (ax as (_, consts), old_weight) =>

                               if exists is_dirty consts then

                                 apfst (cons (ax, NONE))

                               else

                                 apsnd (cons (ax, old_weight)))

                  |>> append (more_rejects

                              |> map (fn (ax as (_, consts), old_weight) =>

                                         (ax, if exists is_dirty consts then NONE

                                              else SOME old_weight)))

               val threshold =

                 1.0 - (1.0 - threshold)

                       * Math.pow (decay, Real.fromInt (length accepts))

               val remaining_max = remaining_max - length accepts

             in

               trace_msg ctxt (fn () => "New or updated constants: " ^

                   commas (rel_const_tab' |> Symtab.dest

                           |> subtract (op =) (rel_const_tab |> Symtab.dest)

                           |> map string_for_hyper_pconst));

               map (fst o fst) accepts @

               (if remaining_max = 0 then

                  []

                else

                  iter (j + 1) remaining_max threshold rel_const_tab'

                       hopeless_rejects hopeful_rejects)

             end

           | relevant candidates rejects

                      (((ax as (((_, stature), _), fact_consts)), cached_weight)

                       :: hopeful) =

             let

               val weight =

                 case cached_weight of

                   SOME w => w

                 | NONE => fact_weight fudge stature const_tab rel_const_tab

                                       chained_const_tab fact_consts

             in

               if weight >= threshold then

                 relevant ((ax, weight) :: candidates) rejects hopeful

               else

                 relevant candidates ((ax, weight) :: rejects) hopeful

             end

         in

           trace_msg ctxt (fn () =>

               "ITERATION " ^ string_of_int j ^ ": current threshold: " ^

               Real.toString threshold ^ ", constants: " ^

               commas (rel_const_tab |> Symtab.dest

                       |> filter (curry (op <>) [] o snd)

                       |> map string_for_hyper_pconst));

           relevant [] [] hopeful

         end

     fun prepend_facts ths accepts =

       ((facts |> filter (member Thm.eq_thm_prop ths o snd)) @

        (accepts |> filter_out (member Thm.eq_thm_prop ths o snd)))

       |> take max_relevant

     fun insert_into_facts accepts [] = accepts

       | insert_into_facts accepts ths =

         let

           val add = facts |> filter (member Thm.eq_thm_prop ths o snd)

           val (bef, after) =

             accepts |> filter_out (member Thm.eq_thm_prop ths o snd)

                     |> take (max_relevant - length add)

                     |> chop special_fact_index

         in bef @ add @ after end

     fun insert_special_facts accepts =

        [] |> could_benefit_from_ext is_built_in_const accepts ? cons @{thm ext}

           |> insert_into_facts accepts

   in

     facts |> map_filter (pair_consts_fact thy is_built_in_const fudge)

           |> iter 0 max_relevant threshold0 goal_const_tab []

           |> not (null add_ths) ? prepend_facts add_ths

           |> insert_special_facts

           |> tap (fn accepts => trace_msg ctxt (fn () =>

                       "Total relevant: " ^ string_of_int (length accepts)))

   end

 (***************************************************************)

 (* Retrieving and filtering lemmas                             *)

 (***************************************************************)

 (*** retrieve lemmas and filter them ***)

 (*Reject theorems with names like "List.filter.filter_list_def" or

   "Accessible_Part.acc.defs", as these are definitions arising from packages.*)

 fun is_package_def a =

   let val names = Long_Name.explode a in

     (length names > 2 andalso not (hd names = "local") andalso

      String.isSuffix "_def" a) orelse String.isSuffix "_defs" a

   end

 fun uniquify xs =

   Termtab.fold (cons o snd)

                (fold (Termtab.update o `(prop_of o snd)) xs Termtab.empty) []

 (* FIXME: put other record thms here, or declare as "no_atp" *)

 fun multi_base_blacklist ctxt ho_atp =

   ["defs", "select_defs", "update_defs", "split", "splits", "split_asm",

    "cases", "ext_cases", "eq.simps", "eq.refl", "nchotomy", "case_cong",

    "weak_case_cong"]

   |> not (ho_atp orelse (Config.get ctxt instantiate_inducts)) ?

         append ["induct", "inducts"]

   |> map (prefix ".")

 val max_lambda_nesting = 3 (*only applies if not ho_atp*)

 fun term_has_too_many_lambdas max (t1 $ t2) =

     exists (term_has_too_many_lambdas max) [t1, t2]

   | term_has_too_many_lambdas max (Abs (_, _, t)) =

     max = 0 orelse term_has_too_many_lambdas (max - 1) t

   | term_has_too_many_lambdas _ _ = false

 (* Don't count nested lambdas at the level of formulas, since they are

    quantifiers. *)

 fun formula_has_too_many_lambdas true _ _ = false (*i.e. ho_atp*)

   | formula_has_too_many_lambdas _ Ts (Abs (_, T, t)) =

       formula_has_too_many_lambdas false (T :: Ts) t

   | formula_has_too_many_lambdas _ Ts t =

     if member (op =) [HOLogic.boolT, propT] (fastype_of1 (Ts, t)) then

       exists (formula_has_too_many_lambdas false Ts) (#2 (strip_comb t))

     else

       term_has_too_many_lambdas max_lambda_nesting t

 (* The max apply depth of any "metis" call in "Metis_Examples" (on 2007-10-31)

    was 11. *)

 val max_apply_depth = 15

 fun apply_depth (f $ t) = Int.max (apply_depth f, apply_depth t + 1)

   | apply_depth (Abs (_, _, t)) = apply_depth t

   | apply_depth _ = 0

 fun is_formula_too_complex ho_atp t =

   apply_depth t > max_apply_depth orelse formula_has_too_many_lambdas ho_atp [] t

 (* FIXME: Extend to "Meson" and "Metis" *)

 val exists_sledgehammer_const =

   exists_Const (fn (s, _) => String.isPrefix sledgehammer_prefix s)

 (* FIXME: make more reliable *)

 val exists_low_level_class_const =

   exists_Const (fn (s, _) =>

      String.isSubstring (Long_Name.separator ^ "class" ^ Long_Name.separator) s)

 fun is_metastrange_theorem th =

   case head_of (concl_of th) of

     Const (s, _) => (s <> @{const_name Trueprop} andalso

                      s <> @{const_name "=="})

   | _ => false

 fun is_that_fact th =

   String.isSuffix (Long_Name.separator ^ Obtain.thatN) (Thm.get_name_hint th)

   andalso exists_subterm (fn Free (s, _) => s = Name.skolem Auto_Bind.thesisN

                            | _ => false) (prop_of th)

 (**** Predicates to detect unwanted facts (prolific or likely to cause

       unsoundness) ****)

 fun is_theorem_bad_for_atps ho_atp exporter thm =

   is_metastrange_theorem thm orelse

   (not exporter andalso

    let val t = prop_of thm in

      is_formula_too_complex ho_atp t orelse exists_type type_has_top_sort t orelse

      exists_sledgehammer_const t orelse exists_low_level_class_const t orelse

      is_that_fact thm

    end)

 fun all_facts ctxt ho_atp reserved exporter add_ths chained_ths css_table =

   let

     val thy = Proof_Context.theory_of ctxt

     val global_facts = Global_Theory.facts_of thy

     val local_facts = Proof_Context.facts_of ctxt

     val named_locals = local_facts |> Facts.dest_static []

     val assms = Assumption.all_assms_of ctxt

     fun is_good_unnamed_local th =

       not (Thm.has_name_hint th) andalso

       forall (fn (_, ths) => not (member Thm.eq_thm_prop ths th)) named_locals

     val unnamed_locals =

       union Thm.eq_thm_prop (Facts.props local_facts) chained_ths

       |> filter is_good_unnamed_local |> map (pair "" o single)

     val full_space =

       Name_Space.merge (Facts.space_of global_facts, Facts.space_of local_facts)

     fun add_facts global foldx facts =

       foldx (fn (name0, ths) =>

         if not exporter andalso name0 <> "" andalso

            forall (not o member Thm.eq_thm_prop add_ths) ths andalso

            (Facts.is_concealed facts name0 orelse

             (not (Config.get ctxt ignore_no_atp) andalso

              is_package_def name0) orelse

             exists (fn s => String.isSuffix s name0)

                    (multi_base_blacklist ctxt ho_atp)) then

           I

         else

           let

             val multi = length ths > 1

             val backquote_thm =

               backquote o string_for_term ctxt o close_form o prop_of

             fun check_thms a =

               case try (Proof_Context.get_thms ctxt) a of

                 NONE => false

               | SOME ths' => eq_list Thm.eq_thm_prop (ths, ths')

           in

             pair 1

             #> fold (fn th => fn (j, (multis, unis)) =>

                         (j + 1,

                          if not (member Thm.eq_thm_prop add_ths th) andalso

                             is_theorem_bad_for_atps ho_atp exporter th then

                            (multis, unis)

                          else

                            let

                              val new =

                                (((fn () =>

                                  if name0 = "" then

                                    th |> backquote_thm

                                  else

                                    [Facts.extern ctxt facts name0,

                                     Name_Space.extern ctxt full_space name0,

                                     name0]

                                    |> find_first check_thms

                                    |> (fn SOME name =>

                                           make_name reserved multi j name

                                         | NONE => "")),

                                 stature_of_thm global assms chained_ths

                                                css_table name0 th), th)

                            in

                              if multi then (new :: multis, unis)

                              else (multis, new :: unis)

                            end)) ths

             #> snd

           end)

   in

     (* The single-name theorems go after the multiple-name ones, so that single

        names are preferred when both are available. *)

     ([], []) |> add_facts false fold local_facts (unnamed_locals @ named_locals)

              |> add_facts true Facts.fold_static global_facts global_facts

              |> op @

   end

 fun external_frees t =

   [] |> Term.add_frees t |> filter_out (can Name.dest_internal o fst)

 fun nearly_all_facts ctxt ho_atp ({add, only, ...} : relevance_override)

                      chained_ths hyp_ts concl_t =

   if only andalso null add then

     []

   else

     let

       val thy = Proof_Context.theory_of ctxt

       val reserved = reserved_isar_keyword_table ()

       val add_ths = Attrib.eval_thms ctxt add

       val ind_stmt =

         (hyp_ts |> filter_out (null o external_frees), concl_t)

         |> Logic.list_implies |> Object_Logic.atomize_term thy

       val ind_stmt_xs = external_frees ind_stmt

       val css_table = clasimpset_rule_table_of ctxt

     in

       (if only then

          maps (map (fn ((name, stature), th) => ((K name, stature), th))

                o fact_from_ref ctxt reserved chained_ths css_table) add

        else

          all_facts ctxt ho_atp reserved false add_ths chained_ths css_table)

       |> Config.get ctxt instantiate_inducts

          ? maps (instantiate_if_induct_rule ctxt ind_stmt ind_stmt_xs)

       |> (not (Config.get ctxt ignore_no_atp) andalso not only)

          ? filter_out (No_ATPs.member ctxt o snd)

       |> uniquify

     end

 fun relevant_facts ctxt (threshold0, threshold1) max_relevant is_built_in_const

                    fudge (override as {only, ...}) chained_ths hyp_ts concl_t

                    facts =

   let

     val thy = Proof_Context.theory_of ctxt

     val decay = Math.pow ((1.0 - threshold1) / (1.0 - threshold0),

                           1.0 / Real.fromInt (max_relevant + 1))

   in

     trace_msg ctxt (fn () => "Considering " ^ string_of_int (length facts) ^

                              " facts");

     (if only orelse threshold1 < 0.0 then

        facts

      else if threshold0 > 1.0 orelse threshold0 > threshold1 orelse

              max_relevant = 0 then

        []

      else

        relevance_filter ctxt threshold0 decay max_relevant is_built_in_const

            fudge override facts (chained_ths |> map prop_of) hyp_ts

            (concl_t |> theory_constify fudge (Context.theory_name thy)))

     |> map (apfst (apfst (fn f => f ())))

   end

 end;